Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 311Let us very briefly recall the origin of the problem. In classical GR,future event horizons behave in a manner that has a peculiar thermodynamicalflavor. This remark, together with a detailed physical analysis ofthe behavior of hot matter in the vicinity of horizons, prompted Bekensteinto suggest that there is entropy associated to every horizon. The suggestionwas first consider ridicule, because it implies that a black hole is hotand radiates. But then Steven Hawking, in a celebrated work [Hawking(1975)], showed that QFT in curved space–time predicts that a black holeemits thermal radiation, precisely at the temperature predicted by Bekenstein,and Bekenstein courageous suggestion was fully vindicated. Sincethen, the entropy of a BH has been indirectly computed in a surprisingvariety of manners, to the point that BH entropy and BH radiance arenow considered almost an established fact by the community, although, ofcourse, they were never observed nor, presumably, they are going to beobserved soon. This confidence, perhaps a bit surprising to outsiders, isrelated to the fact thermodynamics is powerful in indicating general propertiesof systems, even if we do not control its microphysics. Many hopethat the Bekenstein–Hawking radiation could play for quantum gravity arole analogous to the role played by the black body radiation for quantummechanics. Thus, indirect arguments indicate that a Schwarzschild BH hasan entropyS = 1 4AG(3.156)The remaining challenge is to derive this formula from first principles [Rovelli(1997)].Later in the book we will continue our exposition of various approachesto quantum gravity.3.10.5 Basics of Morse and (Co)Bordism Theories3.10.5.1 Morse Theory on Smooth ManifoldsAt the same time the variational formulae were discovered, a related technique,called Morse theory, was introduced into Riemannian geometry. Thistheory was developed by Morse, first for functions on manifolds in 1925,and then in 1934, for the loop space. The latter theory, as we shall see, setsup a very nice connection between the first and second variation formulaefrom the previous section and the topology of M. It is this relationship thatwe shall explore at a general level here. In section 5 we shall then see how